bessel_i

kyosu::bessel_i = eve::functor<bessel_i_t>

inlineconstexpr

Computes the spherical or cylindrical modified Bessel functions of the second kind, extended to the complex plane and cayley_dickson algebras.

#include <kyosu/functions.hpp>

Callable Signatures

namespace kyosu
{
   template<eve;scalar_value N, kyosu::concepts::cayley_dickson_like T>    constexpr auto bessel_i(N n, T z)                                 noexcept; //1
   template<eve;scalar_value N, kyosu::concepts::complex_like T, size_t S> constexpr auto bessel_i(N n, T z, std::span<Z, S> cis)            noexcept; //2
   template<eve;scalar_value N, kyosu::concepts::cayley_dickson_like T>    constexpr auto bessel_i[spherical](N n, T z)                      noexcept; //3
   template<eve;scalar_value N, kyosu::concepts::complex_like T, size_t S> constexpr auto bessel_i[spherical](N n, T z, std::span<Z, S> sis) noexcept; //4
}

Parameters

• n: scalar order (integral or floating)
• z: Value to process.
• sis, cis : std::span of T

Return value

1. returns \(I_n\)(z) (cylindrical).
2. Same as 1, but cis is filled by the lesser orders.
3. returns \(i_n\)(z) (spherical).
4. Same as 3, but sis is filled by the lesser orders.

Note

• Let \( i = \lfloor |n| \rfloor \) and \( j=|n|-i\). If \(n\) is positive the lesser order are \((\pm j, \pm(j+1), \dots, \pm(j+i)) \) with \(+\) sign if \(n\) is positive and \(-\) sign if \(n\) is negative.
• The span parameters are filled according the minimum of their allocated size and \(i\).
• cylindical option can be used and its result is identical to the regular call (no option).

External references

• Wolfram MathWorld: Modified Bessel Function of the First Kind
• Wolfram MathWorld: Spherical Modified Bessel Function of the First Kind
• Wikipedia: Bessel function
• DLMF: Modified Bessel functions

Example